The Riemann hypothesis

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Separation of the complex zeros of the Riemann zeta function

Herman te Riele, CWI Amsterdam

Symposium

Computing by the Numbers: Algorithms, Precision, and Complexity in honor of the 60th birthday of Richard Brent

Berlin, 20–21 July 2006

– p.1/22

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Outline

The Riemann hypothesis History

Some “zeros” of

with Mathematica

From complex to real: the Rieman-Siegel function

Total number of zeros in a given part of the critical strip The Euler-Maclaurin formule voor

The Riemann-Siegel formula for

Gram points and Gram’s “Law”

Rosser’s rule

Separation of zeros of

for Stopping at , knowing zeros The sign of

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The Riemann hypothesis

The Riemann zeta function

is the analytic function of

defined by:

for , and by analytic continuation for . Apart from “trivial” zeros at the negative even integers, all zeros of

lie in the so-called critical strip .

The Riemann hypothesis is the conjecture that all nontrivial zeros of

lie on the so-called critical line

.

– p.3/22

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History

1903 15 Gram

1914 79 Backlund

1925 138 Hutchinson

1935 1,041 Titchmarsh

1953 1,104 Turing

1956 25,000 Lehmer

1958 35,337 Meller

1966 250,000 Lehman

1968 3,502,500 Rosser, Yohe, Schoenfeld

1979 81,000,001 Brent

1982 200,000,001 Brent, Van de Lune, Te Riele, Winter 1983 300,000,001 Van de Lune, Te Riele

1986 1,500,000,001 Van de Lune, Te Riele, Winter

2001 10,000,000,000 Van de Lune

2004 900,000,000,000 Wedeniwski

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Some “zeros” of

with Mathematica

In[39]:= FindRoot@Zeta@sD 0, 8s, 0<D

Out[39]= 8s ® -2.<

8s ® -1.9999999999999973‘<

In[40]:= FindRoot@Zeta@sD 0, 8s, 0.4+ 12 I<D

Out[40]= 8s ® 0.5+ 14.1347ä<

8s ® 0.5000000000000071‘+ 14.134725141734693‘ä<

– p.5/22

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From complex to real: the Rieman- Siegel function

satisfies a functional equation which may be written in the form

with

It follows that, if

then

is real (for real ). Since

, the zeros of are the imaginary parts of the zeros of

on the critical line.

is known as the Riemann-Siegel function.

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Some plots of

with Mathematica

In[63]:= Plot@RiemannSiegelZ@tD,8t, 0, 50<D

10 20 30 40 50

-3 -2 -1 1 2 3

Out[63]= Graphics In[64]:= Plot@RiemannSiegelZ@tD,8t, 50, 100<D

60 70 80 90 100

-4 -2 2 4

Out[64]= Graphics

– p.7/22

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Some plots of with Mathematica

In[63]:= Plot@RiemannSiegelZ@tD,8t, 0, 50<D

10 20 30 40 50

-3 -2 -1 1 2 3

Out[63]= Graphics

In[64]:= Plot@RiemannSiegelZ@tD,8t, 50, 100<D

60 70 80 90 100

-4 -2 2 4

Out[64]= Graphics

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Another plot of

Close zeros: Lehmer phenomenon

7003 7004 7005 7006 7007

-6 -4 -2 2 4

Out[61]= Graphics In[62]:= Plot@RiemannSiegelZ@tD,8t, 7005, 7005.15<,

Ticks®887005.03, 7005.08, 7005.13<, Automatic<D

7005.08 7005.13

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01

Out[62]= Graphics

– p.8/22

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Another plot of

Close zeros: Lehmer phenomenon

7003 7004 7005 7006 7007

-6 -4 -2 2 4

Out[61]= Graphics

In[62]:= Plot@RiemannSiegelZ@tD,8t, 7005, 7005.15<, Ticks®887005.03, 7005.08, 7005.13<, Automatic<D

7005.08 7005.13

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01

Out[62]= Graphics

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Total number of zeros in a given part of the critical strip

Backlund proved around 1912 that the total number

of complex zeros of

with

(counting multiplicities) can be expressed as

where is the broken line segment from (for some )

to to

.

Backlund observed that if on , then this formula suffices to determine

as the nearest integer to

. Expansion:

as

Example:

. _{– p.9/22}

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The Euler-Maclaurin formule for

where

for all , en

.

are the Bernoulli-numbers:

.

For

a good choice for is: and

. Complexity:

.

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The Riemann-Siegel formula for

Let

,

, and

, then

with

and

. For ( ) we have

with

, , and (Doctor’s Thesis Gabcke). Complexity:

.

– p.11/22

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Gram points and Gram’s “Law”

The Gram point is the unique solution in

of the equation

The value of the first term

in the Riemann-Siegel formula in the Gram point equals

Gram’s “Law” is the tendency of the zeros of

to alternate with the Gram points (based on the expectation that the first term in the RS formula dominates).

Unfortunately, this “Law” fails infinitely often, but it is known that on average there is precisely one zero of

inbetween two

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Illustration of Gram’s “Law”

In[88]:= Plot@RiemannSiegelZ@tD,8t, 0, 40<,

Ticks®889.7, 17.8, 23.2, 27.7, 31.7, 35.5, 39.0, 42.34<, Automatic<D

9.7 17.8 23.227.731.735.539 -1

1 2 3

Out[88]= Graphics

– p.13/22

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First violation of Gram’s “Law”

In[112]:=

Plot@RiemannSiegelZ@tD, 8t, 280.8, 284.2<, Ticks®88280.8, 282.45, 284.11<, Automatic<D

282.45 284.11

-7 -6 -5 -4 -3 -2 -1

Out[112]=

Graphics

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Rosser’s Rule

A Gram point is called good if

, and bad otherwise.

A Gram block is an interval

, such that both and are good Gram points whereas the intermediate points

are bad points.

Rosser’s Rule: Gram blocks of length contain exactly zeros.

Richard Brent found the first exception to Rosser’s Rule, namely the Gram block of length 2:

with . This

contains no zeros but it is followed by the Gram block

with three zeros.

– p.15/22

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Example: Sign pattern of on Gram block of length 8

Let be an even (w.l.o.g.) positive integer.

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First violation of Rosser’s Rule

In[8]:= Plot@RiemannSiegelZ@t+6820050D,8t, 0.984896665, 2.341223716<, Ticks®880.985, 1.437, 1.889, 2.341<, Automatic<D

1.437 1.889 2.341

-50 -40 -30 -20 -10

Out[8]= Graphics

In[10]:= Plot@RiemannSiegelZ@t+6820050D,8t, 1.437005684, 2.341223716<, Ticks®881.437, 1.889, 2.341<, Automatic<D

1.889 2.341

-0.3 -0.2 -0.1 0.1 0.2

Out[10]= Graphics

– p.17/22

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Separation of zeros of for

Violations of Rosser’s Rule are extremely rare, so Rosser’s Rule is a very useful heuristic to verify the Riemann hypothesis:

suppose that sign changes of

have been found in

, then evaluate

until the next good Gram point (following ) is found. In roughly 70% of the cases this is , i.e., .

and

then have differ- ent signs, so we have proved the existence of at least one zero between and . The process is continued then with replaced by .

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Separation, the "missing two" zeros

If then we have sign changes on

and we try to find the "missing two" zeros on this Gram block.

If we succeed, then is replaced by and the process is continued; if after a large number of -evaluations our program does not succeed, then either:

1 the block

does contain a pair of (not-yet-found) very close zeros –and the precision of the computations must be increased to find them–, or

2 the "missing two" zeros are found in the preceding or following Gram block (this occurs only roughly 30 times in every million consecutive Gram points), or

3 ???!!!.

A plot of

on

and neighbouring Gram blocks usually reveals where the "missing two" zeros are to be found.

– p.19/22

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Stopping at , knowing

zeros

Theorem (Littlewood, Turing, Lehman, Brent) If consecutive Gram blocks with union

satisfy Rosser’s Rule, where

then

and

Gourdon, verifying RH until zero #

, has

with

,

, and

so was sufficient for proving that

.

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The sign of

Applying backward error analysis to the evaluation of the

Riemann-Siegel sum, Brent gave the following rigorous bound for the error in the computed value

of

with a fast, single-precision method:

where

.

For the range for which Brent verified RH (75,000,000 zeros), the right-hand side is . Brent gave a similar, but much more accurate, bound for a slower, double-precision method which was invoked when the fast method could not determine the sign of

with certainty.

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